Communications in Number Theory and Physics

Volume 3 (2009)

Number 1

Feynman motives of banana graphs

Pages: 1 – 57

DOI: https://dx.doi.org/10.4310/CNTP.2009.v3.n1.a1

Authors

Paolo Aluffi (Department of Mathematics, Florida State University, Tallahassee, Florida, U.S.A.)

Matilde Marcolli (Max–Planck Institut für Mathematik,Bonn, Germany)

Abstract

We consider the infinite family of Feynman graphs known as the “banana graphs” and compute explicitly the classes of the corresponding graph hypersurfaces in the Grothendieck ring of varieties as well as their Chern–Schwartz–MacPherson classes, using the classical Cremona transformation and the dual graph, and a blowup formula for characteristic classes. We outline the interesting similarities between these operations and we give formulae for cones obtained by simple operations on graphs. We formulate a positivity conjecture for characteristic classes of graph hypersurfaces and discuss briefly the effect of passing to noncommutative spacetime.

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Published 1 January 2009